Angle Relationships & Polygons - Accuplacer Qas

The measure of an angle's supplement is 10 degrees less than three times the measure of its complement. What is the measure of the angle?
Answer: 40° Let the angle be $x$ . Its supplement is $180 - x$ and its complement is $90 - x$ . The problem states that the supplement is 10 less than three times the complement. This translates to the equation: $180 - x = 3(90 - x) - 10$ . Distribute the 3: $180 - x = 270 - 3x - 10$ . Simplify: $180 - x = 260 - 3x$ . Add $3x$ to both sides: $180 + 2x = 260$ . Subtract 180 from both sides: $2x = 80$ . Solve for $x$ : $x = 40°$ .
A regular octagon has an interior angle that measures x°. What is the measure of each exterior angle of this octagon?
Answer: 45° Each exterior angle of any regular polygon measures 360° ÷ n, where n is the number of sides. For an octagon, each exterior angle = 360° ÷ 8 = 45°. Choice B (135°) is the measure of each interior angle of a regular octagon. Choice C (225°) is 45° × 5, possibly from miscounting or confusion. Choice D (315°) is 360° - 45°, representing a reflex angle interpretation.
The sum of the interior angles of a polygon is 1980°. How many sides does this polygon have?
Answer: 13 The sum of interior angles of an n-sided polygon is (n-2) × 180°. Setting up the equation: (n-2) × 180° = 1980°. Dividing both sides by 180°: n-2 = 11, so n = 13. Choice A (11) is the value of (n-2). Choice C (15) results from dividing 1980 by 132 instead of 180. Choice D (17) results from adding 4 to 13 instead of 2.
In a right triangle, the measures of the two acute angles are $(7x + 2)°$ and $(3x - 2)°$ . What is the measure of the smaller acute angle?
Answer: 25° The two acute angles in a right triangle are complementary, meaning their sum is 90°. Set up the equation: $(7x + 2) + (3x - 2) = 90$ . Combine like terms: $10x = 90$ . Solve for $x$ : $x = 9$ . Now substitute $x = 9$ into both expressions to find the angle measures. The first angle is $7(9) + 2 = 63 + 2 = 65°$ . The second angle is $3(9) - 2 = 27 - 2 = 25°$ . The smaller of the two angles is 25°.
Two lines intersect, forming four angles. One of the angles measures 42°. What is the sum of the measures of the two angles that are adjacent to this 42° angle?
Answer: 276° When two lines intersect, adjacent angles are supplementary (add up to 180°). The 42° angle has two adjacent angles. The measure of the first adjacent angle is $180° - 42° = 138°$ . The second adjacent angle is on the other side and is also supplementary to the 42° angle, so its measure is also $180° - 42° = 138°$ . The sum of the measures of these two adjacent angles is $138° + 138° = 276°$ .
The exterior angle of a regular polygon is 1/8 of its interior angle. How many sides does the polygon have?
Answer: 18 Let the exterior angle be $x$ and the interior angle be $y$ . We are given $x = \frac{1}{8}y$ , which means $y = 8x$ . The interior and exterior angles at a vertex are supplementary, so $x + y = 180°$ . Substitute $8x$ for $y$ : $x + 8x = 180°$ , which simplifies to $9x = 180°$ . Solving for $x$ gives $x = 20°$ . The number of sides $n$ of a regular polygon is $360°$ divided by the measure of one exterior angle. So, $n = 360 / 20 = 18$ . The polygon has 18 sides.
In triangle XYZ, the measure of angle X is 10 degrees greater than the measure of angle Y, and the measure of angle Z is twice the measure of angle Y. What is the measure of the largest angle in the triangle?
Answer: 85° Let the measure of angle Y be $y$ . Then the measure of angle X is $y + 10$ , and the measure of angle Z is $2y$ . The sum of the angles in a triangle is 180°. So, $(y + 10) + y + 2y = 180$ . Combine like terms: $4y + 10 = 180$ . Subtract 10 from both sides: $4y = 170$ . Divide by 4: $y = 42.5°$ . Now find the measures of all three angles: Angle Y = 42.5°; Angle X = 42.5° + 10° = 52.5°; Angle Z = 2 * 42.5° = 85°. The largest angle is angle Z, which is 85°.
In triangle ABC, the exterior angle at vertex C measures 118°. If angle A measures 42°, what is the measure of angle B?
Answer: 76° When you encounter a triangle problem involving an exterior angle, you're working with one of geometry's most useful relationships: an exterior angle of a triangle equals the sum of the two non-adjacent interior angles. Here, the exterior angle at vertex C measures 118°, and angle A measures 42°. Since the exterior angle at C is formed by extending one side of the triangle, it equals the sum of the two remote interior angles (A and B). So: $118° = 42° + \text{angle B}$ Solving for angle B: $\text{angle B} = 118° - 42° = 76°$ Let's examine why the other answers miss the mark. Choice A (34°) would result if you mistakenly subtracted the exterior angle from angle A ( $42° - 118°$ , then took the absolute value), which reverses the relationship. Choice B (62°) appears if you incorrectly assume the exterior angle plus angle A should equal 180° and solve $180° - 118° = 62°$ , but this confuses the exterior angle theorem with linear pair relationships. Choice D (138°) would make the sum of all interior angles exceed 180° ( $42° + 138° = 180°$ , leaving 0° for angle C), violating the triangle angle sum theorem. You can verify: if angle B = 76°, then angle C = $180° - 42° - 76° = 62°$ , and the exterior angle should equal $42° + 76° = 118°$ ✓ Remember this key relationship: exterior angle = sum of the two remote interior angles. It's often faster than finding all three interior angles first.
In quadrilateral PQRS, consecutive angles measure 85°, 110°, and 95°. If the quadrilateral is cyclic (inscribed in a circle), what must be true about the fourth angle?
Answer: It measures 70° and is opposite the 110° angle In a cyclic quadrilateral, opposite angles are supplementary. The fourth angle must be 360° - (85° + 110° + 95°) = 70°. Since consecutive angles are given as 85°, 110°, 95°, the fourth angle (70°) is opposite to 110°. Choice B incorrectly identifies which angle is opposite. Choices C and D give incorrect angle measures that don't satisfy the angle sum property.
In isosceles triangle ABC, side AB is congruent to side AC. A line parallel to base BC intersects side AB at D and side AC at E. If the measure of the vertex angle BAC is 30°, what is the measure of angle ADE?
Answer: 75° First, find the measure of the base angles in the large isosceles triangle ABC. The sum of angles is 180°. The sum of the two equal base angles (ABC and ACB) is $180° - 30° = 150°$ . Each base angle is $150° / 2 = 75°$ . So, angle ABC = 75°. Because the line segment DE is parallel to the base BC, the angle ADE and the angle ABC are corresponding angles. Therefore, their measures are equal. Angle ADE = 75°.
The measures of the four interior angles of a quadrilateral are in the ratio 1:3:5:6. What is the measure of the smallest angle?
Answer: 24° The sum of the interior angles of a quadrilateral is 360°. Let the angles be $x$ , $3x$ , $5x$ , and $6x$ . Their sum is $x + 3x + 5x + 6x = 15x$ . Set this sum equal to 360°: $15x = 360°$ . Divide by 15 to find $x = 24°$ . The smallest angle is represented by $x$ , so its measure is 24°.
Two angles are supplementary. One angle is 30 degrees less than twice the other angle. What is the measure of the smaller angle?
Answer: 70° Let the two angles be $x$ and $y$ . Since they are supplementary, $x + y = 180$ . One angle is 30 less than twice the other, so we can write $y = 2x - 30$ . Substitute the second equation into the first: $x + (2x - 30) = 180$ . Combine like terms: $3x - 30 = 180$ . Add 30 to both sides: $3x = 210$ . Divide by 3: $x = 70$ . Now find the other angle: $y = 180 - 70 = 110$ . The two angles are 70° and 110°. The smaller angle is 70°.
A four-sided plot of land is a convex quadrilateral. A surveyor measures three of the interior angles to be 85°, 92°, and 78°. To ensure the survey is accurate for a flat, four-sided figure, what must the measure of the fourth angle be?
Answer: 105° The sum of the interior angles of any convex quadrilateral (a four-sided polygon) is always 360°. To find the measure of the fourth angle, first sum the three known angles: $85° + 92° + 78° = 255°$ . Then, subtract this sum from the total of 360°: $360° - 255° = 105°$ . The fourth angle must be 105°.
Two parallel lines are cut by a transversal. If one of the alternate interior angles measures (3x + 15)° and its corresponding alternate interior angle measures (5x - 25)°, what is the value of x?
Answer: 20 Alternate interior angles are equal when parallel lines are cut by a transversal. Setting (3x + 15) = (5x - 25), we get 15 + 25 = 5x - 3x, so 40 = 2x, giving x = 20. Choice A (10) results from solving 3x + 15 + 5x - 25 = 180 (treating them as supplementary). Choice C (25) comes from setting 3x + 15 = 90. Choice D (35) results from solving 3x + 15 = 5x - 25 incorrectly as 15 + 25 = 5x + 3x.
Four of the interior angles of a pentagon measure 90°, 100°, 105°, and 115°. What is the measure of the fifth angle?
Answer: 130° First, determine the sum of the interior angles of a pentagon (a polygon with 5 sides) using the formula $(n-2) \times 180°$ . For a pentagon, this is $(5-2) \times 180° = 3 \times 180° = 540°$ . Next, sum the four given angles: $90° + 100° + 105° + 115° = 410°$ . To find the fifth angle, subtract this sum from the total sum of the angles: $540° - 410° = 130°$ .
A regular pentagon is inscribed in a circle. If two adjacent vertices of the pentagon are connected to the center of the circle, what is the measure of the angle formed at the center?
Answer: 72° When you see a regular polygon inscribed in a circle, you're dealing with central angles that divide the circle into equal parts. A regular pentagon has 5 equal sides and 5 equal vertices, so the circle is divided into 5 congruent central angles. Since a complete circle measures 360°, each central angle in a regular pentagon measures $\frac{360°}{5} = 72°$ . When you connect two adjacent vertices to the center, you're creating exactly one of these central angles. Let's examine why the other answers are incorrect. Choice A (36°) represents half of the correct central angle - this might come from incorrectly thinking about the relationship between central and inscribed angles, or from confusing this with interior angle calculations. Choice B (108°) is the measure of each interior angle of a regular pentagon, which is calculated using the formula $\frac{(n-2) \times 180°}{n}$ , but this isn't what the question asks for. Choice D (144°) equals two central angles ( $2 \times 72° = 144°$ ), which would be the angle if you connected vertices that are separated by one vertex, not adjacent ones. The correct answer is C (72°) because it represents one-fifth of the complete 360° circle. Strategy tip: For any regular n-sided polygon inscribed in a circle, the central angle between adjacent vertices is always $\frac{360°}{n}$ . Memorize this formula - it's a reliable shortcut that works for any regular polygon, from triangles to dodecagons.
In triangle ABC, the measure of the exterior angle at vertex C is 110°. The measure of the interior angle at vertex B is $(2x + 10)°$ , and the measure of the interior angle at vertex A is $(3x)°$ . What is the measure of angle A?
Answer: 60° The exterior angle theorem states that the measure of an exterior angle of a triangle is equal to the sum of the measures of its two remote interior angles. In this case, the exterior angle at C is equal to the sum of the interior angles at A and B. So, $3x + (2x + 10) = 110$ . Combine like terms: $5x + 10 = 110$ . Subtract 10 from both sides: $5x = 100$ . Divide by 5: $x = 20$ . The question asks for the measure of angle A, which is $3x$ . So, angle A = $3(20) = 60°$ .
Parallel lines $l$ and $m$ are intersected by a transversal line $t$ . The measure of an angle is given by $(3x + 10)°$ . The measure of its alternate interior angle is $(5x - 30)°$ . What is the measure of a consecutive interior angle to the first angle?
Answer: 110° Alternate interior angles are equal. So, set the expressions equal to each other: $3x + 10 = 5x - 30$ . Solve for $x$ : subtract $3x$ from both sides to get $10 = 2x - 30$ . Add 30 to both sides: $40 = 2x$ . Divide by 2: $x = 20$ . Now find the measure of the first angle: $3(20) + 10 = 60 + 10 = 70°$ . Consecutive interior angles are supplementary, meaning they add up to 180°. So, the measure of the required angle is $180° - 70° = 110°$ .
If the number of sides of a regular polygon is doubled, what is the effect on the measure of each exterior angle?
Answer: It is halved. The measure of one exterior angle of a regular polygon with $n$ sides is given by the formula $360°/n$ . If the number of sides is doubled to $2n$ , the new exterior angle measure will be $360°/(2n)$ , which simplifies to $\frac{1}{2} \times (360°/n)$ . This is exactly half of the original exterior angle measure. Therefore, the measure of each exterior angle is halved.

Start this quiz